Extremal Polynomials with Preassigned Zeros and Rational Approximants

Abstract

Let pn be the n th orthonormal polynomial with respect to a positive finite measure μ supported by Δ=[-1,1] . It is well known that, uniformly on compact subsets of C/Δ , \(\liminf_{n\to\infty}|p_n(z)|^{1/n}\ge e^{g_\Omega(z)}\) and, for a large class of measures μ , \(\lim_{n\to\infty}|p_n(z)|^{1/n}=e^{g_\Omega(z)},\) where gΩ(z) is Green's function of \(\Omega=\overline{{\bf C}}\backslash \Delta\) with pole at infinity. It is also well known that these limit relations give convergence of the diagonal Pade approximants of the Markov function \(f(z)=\int_{-1}^1{d\mu(t)\over z-t}\) to f on Ω with a certain geometric speed measured by gΩ(z) .